Daniel's triple challenge

Algebra Level 5

How many ordered triples of complex numbers $(a, b, c)$ are there such that $a^3-b$, $b^3-c$, and $c^3-a$ are rational numbers, and

$a^2(a^4+1)+b^2(b^4+1)+c^2(c^4+1)=2(a^3b+b^3c+c^3a)?$

This problem is posed by Daniel C.

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